Two Generalizations of the Binomial Distribution

SUMMARY The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two-parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second- or higher-order interactions. Two distinct generalizations are obtained, depending on whether the or additive definition of interaction for discrete variables is used. The multiplicative generalization gives rise to a two-parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta-binomially distributed variable the variance always exceeds the corresponding binomial variance, the additive or generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica.